Einstein's Theory of General Relativity

Edited by Oscar Haase, Fall 2022

Einstein's Theory of General Relativity is our current best model of gravity. It expands on the ideas of special relativity in a way that accounts for an apparent paradox when trying to consolidate special relativity with Newtonian gravity. Newton's theory has no description of the method through which gravitation interactions arise, and as such, seems to suggest that gravity is instantaneous "action at a distance". Special relativity asserts that the speed of light is the ultimate speed limit in the universe, and as such, gravity should not be able to surpass that. Because of this contradiction, a new theory of gravity was necessary.

Since its conception in 1915, multiple observations of the theory have been tested and experimentally verified, proving general relativity to be a remarkably successful theory. As well as this, new predictions have been made through solving the equations in this theory and subsequently observed in reality.

The Foundations of General Relativity

As described above, general relativity came about as a means of consolidating special relativity with Newtonian gravity. Those two theories are among a few core ideas that form the basis of Einstein's general theory of gravity.

Special Relativity

The concepts of special relativity are fundamental to the development of general relativity. Special relativity can be thought of as the "special" case of general relativity, wherein objects are not accelerating, assumed to have small masses, and little to no gravitational influence. However, its core ideas still hold:

• Light is the absolute speed limit of the universe, [math]\displaystyle{ c }[/math]
• Light maintains the same speed in every reference frame
• There is an equivalence between mass and energy, [math]\displaystyle{ E = mc^2 }[/math]

Newtonian Gravity

Despite superseding it, general relativity uses Newtonian gravity as a foundation. Newton's theory was incredibly successful, and accurately described the phenomenon of gravity in all but the most extreme cases. As such, it was necessary for general relativity to return to Newtonian gravity in the limiting cases of weak gravity and slow speeds. This proved to be a fundamental part of the theory of general relativity, and leads to Newton's gravitational constant [math]\displaystyle{ G }[/math] appearing in the Einstein field equations.

The Equivalence Principle

The equivalence principle is something that may seem to the introductory physics student as obvious, but at a deeper level is a profound idea that, together with special relativity, leads directly to the theory of general relativity. The equivalence principle states that the mass that keeps things from moving (i.e. inertial mass) is the same as the mass that causes the attraction between two massive bodies (i.e. gravitational mass). In terms of math, this states that the mass present in Newton's second law:

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A visual representation of the equivalence principle

[math]\displaystyle{ \vec{F_i} = m_i \vec{a} }[/math]

is equivalent to the mass present in Newton's law of gravtitation:

[math]\displaystyle{ \vec{F_g} = G \frac{m_g M}{r^2} \hat{r} }[/math]

Note here that the two masses we are discussing are [math]\displaystyle{ m_i }[/math] and [math]\displaystyle{ m_g }[/math], whereas [math]\displaystyle{ M }[/math] is the mass of the object that is attracting the object of interest. Equating these two forces and rearranging yields:

[math]\displaystyle{ m_i \vec{a} = G \frac{m_g M}{r^2} \hat{r} \Rightarrow \vec{a} = \frac{m_i}{m_g}\vec{g} }[/math]

In principle, there's no reason [math]\displaystyle{ \frac{m_i}{m_g} }[/math] should be equal to one. However, experiments have shown to high precision that this is the case. The most notable implication of this is that it is impossible for a confined observer to know whether they are being accelerated uniformly by say, a spaceship, or are merely inside of a uniform gravitational field. As such, we can apply the principles of special relativity to an observer in a uniform gravitational field by thinking of it as an accelerating frame. This leads to the idea that light will bend in the presence of a gravitational field.

Spacetime as a Manifold

From special relativity, we know that the speed of light is constant. So, if light is bending, the only possible explanation is for the two things that make up speed- distance and time- to be bending as well. They must bend inversely in such a way so that the total speed of light is constant. (Note: this is a very hand wavey explanation. However, it does have merit in making the concept of curved spacetime more intuitive to understand.) This led Einstein to combine our three spatial dimensions and single time dimension into four spacetime dimensions, and postulate that matter and energy actually cause curvature in this spacetime.

This curvature is explained using the math of differential geometry, which utilizes tensor calculus to describe manifolds and, more importantly, the paths one can take along such a manifold (for our purposes, a manifold can be thought of as simply a curved surface that looks locally flat. Note that this surface could exist in any dimension, such as our four dimensional spacetime).

Mathematical Framework

Einstein developed a generalized coordinate system and summation notation to simplify his work and create a much more elegant system to describe his ideas. There are four important quantities to understand before tackling the Einstein Field Equations: the metric tensor, Christoffel symbols, geodesic equations, and the Reimann tensor.

The Metric Tensor

The metric tensor is a very important mathematical object in general relativity. Much of the information that describes a curved space is encoded in this object.

A tensor is a multidimensional quantity that describes direction and magnitude in a much more detailed way than a vector. For example, stress in an object is complex and contains many directions of forces at one single point, but by using a stress tensor one may compactly describe a point or even a collection of points. Writing equations in terms of tensors provides a very important quality: a tensor equation that equals zero in one frame of reference will equal zero in all frames of reference. This property provides a means for the study of physical phenomena in any system of coordinates imaginable.

The metric tensor describes distances in a curved four dimensional space. Essentially, it is a matrix whose components correspond to the dot products of each combination of a particular set of unit vectors that span the space. For example, in flat cartesian space:

[math]\displaystyle{ g_{uv} = \begin{bmatrix} \hat{x} \cdot \hat{x} & \hat{x} \cdot \hat{y} & \hat{x} \cdot \hat{z} \\ \hat{y} \cdot \hat{y} & \hat{y} \cdot \hat{y} & \hat{y} \cdot \hat{z} \\ \hat{z} \cdot \hat{x} & \hat{z} \cdot \hat{y} & \hat{z} \cdot \hat{z} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} }[/math]

For this metric tensor [math]\displaystyle{ g_{uv} }[/math], [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are both indices, having values of 0, 1, and 2 (or, [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], and [math]\displaystyle{ z }[/math]). Think of them as corresponding to the row and column coordinate. Flat space has a very familiar metric tensor; the identity matrix. This corresponds to the Euclidean geometry we are all very familiar with.

An important distinction is that this matrix is only a coordinate representation of the metric tensor. Tensors, in a similar way we are familiar with dealing with vectors (in fact, vectors are tensors), can be expressed in different coordinate systems. For example, it's possible to represent the same three dimensional flat space with a different set of coordinates- say, spherical coordinates:

[math]\displaystyle{ g_{uv} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2{\theta} \\ \end{bmatrix} }[/math]

Notice here that, instead of [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], and [math]\displaystyle{ z }[/math], [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] correspond to [math]\displaystyle{ r }[/math], [math]\displaystyle{ \theta }[/math], and [math]\displaystyle{ \phi }[/math].

Spacetime, on the other hand, has four dimensions. As such, we add a fourth coordinate, time, and a corresponding unit vector:

[math]\displaystyle{ g_{\mu \nu} = \begin{bmatrix} \hat{t} \cdot \hat{t} & \hat{t} \cdot \hat{x} & \hat{t} \cdot \hat{y} & \hat{t} \cdot \hat{z} \\ \hat{x} \cdot \hat{t} & \hat{x} \cdot \hat{x} & \hat{x} \cdot \hat{y} & \hat{x} \cdot \hat{z} \\ \hat{y} \cdot \hat{t} & \hat{y} \cdot \hat{x} & \hat{y} \cdot \hat{y} & \hat{y} \cdot \hat{z} \\ \hat{z} \cdot \hat{t} & \hat{z} \cdot \hat{x} & \hat{z} \cdot \hat{y} & \hat{z} \cdot \hat{z} \\ \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} }[/math]

Notice that the dot product of the time basis vectors is negative. This is necessary to keep the speed of light constant (recall above where we described space and time as having to bend in opposite ways).

Curved space comes about when the components of the metric tensor depend on position. We've already seen the spherical metric tensor, which showcases this phenomena. The coordinate dependency of the metric tensor components in the spherical metric tensor corresponds to coordinate curvature, wherein the coordinates we are using cause some curvature in the metric while the physical space it's describing remains flat. General relativity on the other hand often deals with scenarios in which the physical space itself is curved. A basic example of a curved space metric tensor is the Schwarzschild metric, which was one of the first metrics to be solved from Einstein's equations. The metric describes a the spacetime around a spherically symmetric mass.

[math]\displaystyle{ g_{\mu \nu} = \begin{bmatrix} -(1-\frac{2M}{r}) & 0 & 0 & 0 \\ 0 & (1-\frac{2M}{r})^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2{\theta} \\ \end{bmatrix} }[/math]

Despite the relative simplicity of the metric, it presents an interesting topic to study. By using the Schwarzschild metric, one may arrive at singularities, or mathematical points that explode to infinity. At these singularities, black holes are created, and these such points are still the subject of intense research.

Metric tensors are the solutions to the Einstein Field Equations. The field equations describe the relationship between the curvature of spacetime and the mass and energy present in it, and as such output a metric describing the resulting spacetime curvature.

Christoffel Symbols

Christoffel symbols can be loosely thought of as a residual when taking the derivative in a nonlinear coordinate system. If the coordinate system itself depends on a set of parameters, then taking the derivative of a function will not result in a simple derivative. Because of the product rule, there remains a correction term that must be required, and such term is the christoffel symbol. With respect to the metric tensor, the christoffel symbol has a concrete description of the tensor, and represents the correction quantity that must be used to describe geodesics.

[math]\displaystyle{ \Gamma^{\alpha}_{\beta \gamma} = \frac{1}{2} g^{\alpha \delta} (\frac{\partial g_{\delta \beta}}{\partial x^{\gamma}} + \frac{\partial g_{\delta \gamma}}{\partial x^{\beta}} - \frac{\partial g_{\beta \gamma}}{\partial x^{\delta}}) }[/math]

Geodesic Equation

The geodesic equation describes the straightest path along a given manifold. For flat space, or simple cartesian coordinates, if a particle moves then it must move in a straight line. However, for curved space, say for example a sphere, the shortest path between to points is actually curved. This is why planes appear to travel in curved trajectories on 2D maps- they are following the shortest path along the three dimensional sphere of the Earth.

In much the same way, objects in our reality will follow the shortest path along the four dimensional manifold of spacetime. To us, this will appear as traveling in a straight line in the absence of any spacetime curvature. But, when a body is present and causing spacetime to curve, the straightest 4D line now appears to curve to us in 3D. This manifests itself as what we call gravity.

The geodesic equation describes this motion as a set of of second order differential equations, akin to Newton's law of acceleration. Within the mathematical framework, the geodesic equation employs the Christoffel symbol to correct for distortions in spacetime:

[math]\displaystyle{ \frac{d^2 x^{\alpha}}{d \tau^2} = - \Gamma^{\alpha}_{\beta \gamma} \frac{d x^{\beta}}{d \tau} \frac{d x^{\gamma}}{d \tau} }[/math]

Reimann Tensor

Although the geodesic equation can describe motion in curved spacetime, the equation itself is insufficient in describing space. For such a description, one must turn to the Reimann Curvature Tensor. This tensor is one of the most common objects used to describe curved manifolds, and it plays a very important role in the Einstein Field Equations. The Einstein tensor itself contains a derivation of the Reimann tensor, the ricci curvature tensor, and is what results from energy and mass tensor quantities.

The tensor is constructed by taking covariant derivatives of the metric:

but it can also be rewritten in terms of the Christoffel symbols:

[math]\displaystyle{ R^{\rho}_{\sigma \mu \nu} = \frac{\partial \Gamma^{\rho}_{\nu \sigma}}{\partial x^{\mu}} - \frac{\partial \Gamma^{\rho}_{\mu \sigma}}{\partial x^{\nu}} + \Gamma^{\rho}_{\mu \lambda} \Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda} \Gamma^{\lambda}_{\mu \sigma} }[/math]

An important idea that can be taken from the Reimann Tensor is parallel transport. Imagine one is on a hill and facing one direction. Move a certain distance, then move in another path, but this time, remain perpendicular to the surface. Repeat the movement until one ends up in the same point one started at. If one is facing a different direction than when one initially started, then there exists curvature inherent in the manifold. Parallel transport provides an effective means through which to describe the curvature of spacetime.

The Einstein Field Equations

The Einstein Field Equations are the governing equations of general relativity. They relate the energy and mass density in a given area of spacetime to the curvature it causes around that spacetime. In the tensor notation we've been using thus far, the field equations are:

[math]\displaystyle{ R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} }[/math]

As we've seen before, [math]\displaystyle{ g_{\mu \nu} }[/math] is the metric tensor, which describes the curvature of spacetime. [math]\displaystyle{ R_{\mu \nu} }[/math] and [math]\displaystyle{ R }[/math] both come from the Reimann tensor, meaning that in the end they are also functions of the metric tensor. [math]\displaystyle{ T_{\mu \nu} }[/math] is the energy-momentum tensor, and describes the amount of energy and mass in a given area- this is what causes the curvature of spacetime. The constant out front, [math]\displaystyle{ \frac{8 \pi G}{c^4} }[/math], comes about to make the field equations equal to Newtonian gravity in a weak field limit. And, finally, [math]\displaystyle{ \Lambda }[/math] is called the "cosmological constant", and accounts for the innate energy density of the universe. Though it was originally present in Einstein's formulation of the equations, he later removed it. However, it was later reintroduced once the expansion of the universe was observed to be accelerating- a concept closely related to dark energy.

Experimental Verifications

Orbit of Mercury

In a classical two-body system, one object orbits another in a predictable manner. However, the observation of Mercury's orbit demonstrated a precession, which can be visualized as the orbit itself rotating around the Sun. It was only until Einstein introduced his theory that the precession was accurately accounted for.

Gravitational Lensing

When light passes through an object, it follows the geodesic trajectory described by Einstein's equations, and as a result bends. The first confirmation of gravitational lensing of light resulted from the measurement of a star's location during a solar eclipse. On May 1919 Arthur Eddington and his team observed stars near the sun and concluded that Einstein's predictions were consistent with empirical results.

Gravitational Redshift

The effect was only accurately measured in 1959 with the Pound-Rebka Experiment.

Connectedness

How is this topic connected to something that you are interested in?

I have always been fascinated by how gravity can be described in a rigorous mathematical sense, and the revolutionary nature of Einstein's work.

How is it connected to your major?

Electrical Engineers, when designing satellites, have to take into account the effects of GR in order to produce accurate time measurements. Recent experiments have also sought to measure minuscule changes in length and time due to gravitational waves and high velocities.

Is there an interesting industrial application?

For now, GR is restricted to mostly space applications. Away from the Earth's gravity, residents or machines orbiting the earth or traveling through space experience different effects on time and space due to fluctuating gravitational fields.

History

Einstein spent nearly 10 years refining his theory, from 1907 to 1915. After he published his results on Special Relativity, Einstein wanted to incorporate gravity into his theory, but did not realize how to do so until he stumbled upon differential geometric methods.

When the theory was first introduced, the empirical evidence for the theory did not exist, and many scientists around the world were eager to test Einstein's theories. Because of the counter-intuitive nature of Einstein's works, many doubted the validity of the theory, but with the the verification of the precession of mercury and gravitational lensing phenomena, relativity was all but confirmed.

Further reading

https://en.wikipedia.org/wiki/General_relativity

References

http://www.physicsclassroom.com/mmedia/momentum/cthoe.cfm

Einstein, Albert. Relativity: The Special and General Theory. Methuen & Co Ltd, 1916. Print.

Pound, R. V.; Rebka, Jr. G. A. (November 1, 1959). "Gravitational Red-Shift in Nuclear Resonance". Physical Review Letters 3 (9): 439–441.

Rosenthal-Schneider, Ilse: Reality and Scientific Truth. Detroit: Wayne State University Press, 1980. p 74